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We defined in the class branched covering as follows.

Let $\Sigma_1, \Sigma_2$ be two surfaces, $f: \Sigma_1 \longrightarrow \Sigma_2$ is a branched covering if $\forall y \in \Sigma_2 $ there exist $V\subset \Sigma_1$ containing y so that $f^{-1}(V)= U_1 \cup U_2 \cup ... \cup U_n $ so that $f: U_j \longrightarrow V$ is given by $z \longrightarrow z^k$ for $k\geq1$. The points which k>1 called ramification points.

Now, I didnot understand the following. When I searched on the web the standard definition for branched covering is like it is a covering except on some small set. So

  1. How are these two definitions related?
  2. As far as I understand, in the definition I give, we view the surface as a complex manifold (with a complex structure on it) but then the question is there might be lots of complex structure which are not biholomorphic to each other, is the definition independent of the complex structure we put on the surface?
  3. I would appreciate if you suggest some source explaining monodromy, branched coverings.
  4. To understand the first definition a little bit, I took the standard sphere in $\mathbb {R}^3= \mathbb{C} \times R $ and the map taking $(z,t) \longrightarrow (z^2,t)$. This is a branched covering; the north and south poles being branched points with respect to the second definition. Now I will try to see that it is a branched covering with these branch points with the first definition. I need to have some complex coordinates on the sphere and must express the map so that hopefully it will be some power of z, so that I can decide what kind of point is that. Am I on the right track?
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  • $\begingroup$ The definition that you wrote is meaningless (since $U_j, V$ are not subsets of the complex plane). Start by defining an ordinary covering map. $\endgroup$ – Moishe Kohan Nov 8 '17 at 1:53

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